Relation between the Power Law Behavior of MSD and Trajectories of Ions

In Fig. 5.8, positions of atoms and ions at 600 К during 8 ns run for lithium ions in lithium disilicate are shown. In this case, the time scale of the observation is longer than t_dif, even though both localized and diffusive motions along jump paths can be observed.

Figure 5.8 Trajectories of Li ions in Li2Si20s glass at 600 К for a slice of the system during 8 ns run. The motion is highly heterogeneous. That is, both localized jumps and diffusive jumps within ion channels are observed.

Figure 5.9 Temperature dependence of characteristic times in MSD for Li ions in lithium metasilicate replotted from the data shown in [17]. The slope of diffusion coefficient is comparable to that for 1 /t_dif. The dotted dash line is for diffusion coefficient, which is arbitrarily shifted vertically for comparison. Larger changes in slopes are found at longer time scales. Activation energy of the l/t_dif is comparable to the diffusive motion. Adapted with permission from Habasaki, J., Ngai, K. L. (2017), Nearly constant loss [NCL) in lithium metasilicate glass at low temperatures—anisotropic and dynamical caging from molecular dynamics simulations, / Phys. Chem. C, 121, pp. 13729-13737. Copyright [2017] American Chemical Society.

Positions of Si and 0 atoms are essentially localized except for some fluctuation. The set of points for Li in the site is separated from the others but tends to show a pattern related to the jump paths observed at higher temperatures or at a longer time scale (see also Fig. 3.4 and Fig. 3.6 in Chapter 3 for examples of alignment and motions of ions). Each trajectory is the complex ones due to correlated motions, while in functions such as van Hove functions, net changes at distance r are measured. One can easily imagine that the localized motion within a cage becomes dominant at low temperatures. From such trajectories, the power-law region of MSD can be characterized using the fractal dimension analysis.

Cooperative and correlated jumps of ions

Deviations from the simple random walks are often found in MD simulations in limited time regions. The motion of ions in ionically conductive ions is highly cooperative as shown in early molecular dynamics works in the 1990s for Ag⁺ ions [50] and Li⁺ ions [51]. Ions tend to show forward correlated motions with long length scales. For the former, it was attributed to Frenkel-type defects. For the latter, it was pointed out that the distribution of displacements has an inverse power law tail (with an exponential truncation) so that the non-Gaussian dynamics related to this distribution are characterized [51] as Levy flight dynamics [52]. In such motions, cooperative jumps play important roles. Similar dynamics are also known for nonionic materials even for the MD simulations of the soft-core system without attractive terms [53].

In some of the early works, cooperative jumps were regarded to be very rare events, so that their contribution was negligible. This is probably because of a short time scale covered by simulations, which can be addressed by molecular dynamics due to the limited power of computers. However, later, it was pointed out that such motion should be the main mechanism in lithium metasilicate glass [51].

Similar cooperative jumps of several units have been observed in colloids [46], by confocal microscopy. This is a useful method for three-dimensional imaging and characterization of relaxation in colloidal systems as reviewed in [54].

Recently, cooperative jumps of ions seem to be "re-discovered” in several fast conductors [55-58]. It is worth to note that the existence of cooperative motions is confirmed in the works using in ab initio MD as well as classical MD and it means that the phenomena are essentially explained by the classical dynamics.

Power law region and coexistence of fast and slow ions

Before the diffusive region region of MSD is reached, the power- law region is found between t_x2 and t_dif. The power-law exponent in this region is correlated well with the complexity of the motion of ions with backward and forward correlated jumps. To characterize the power law exponent in this region, fractal dimension of random walk, d_w, can be used (see Chapter 4). The value is a measure of the complexity of the motion of ions (or other units). The dimension defined by the following equation can be easily obtained from the trajectories of ions in MD:

where L is a length of the divider and N_T is the number of procedures using a divider to cover the trajectories. Thus, the value can be determined from the slope of the N against L in a double logarithmic plot.

For the free random walk, one can expect that d_w = 2. The values of d_w larger than 3 mean that the trajectories are folded many times by the backward correlation and therefore the motion is localized and slowed down. On the other hand, a strong forward correlation reduces the values of d_w

As known in the theory of fractal for the walks of particles, the power law dependence ~1? of the MSD, is characterized by d_w from the relation (5.10) [4], when the geometrical term governs the slope and if the motion is a simple walk. For example, the power law exponent of MSD was shown to be comparable to the 2/d_w along with the composition dependence of the mixed alkali effect in ion dynamics, while in single alkali systems, one needs to consider contributions of motions with different length scales.

Thus, the analysis of the fractal dimension of the random walks is useful to examine the dynamical changes in the modification of the system.

Diffusive Region of MSD and Exchange of Fast and Slow Motion in Longer Time Scales

From the slope after t_dif, the diffusion coefficient, D, of ions can be determined by the following Einstein equation [59]:

where D is the diffusion coefficient and r, is the positional vector.

This relation holds when the path dynamically formed is percolated when t_dif is attained. Therefore, the calculation of this coefficient should be done after t_dif.

Sometimes, the time region between t_xl and t_x2 might be regarded as the diffusive region because of proportionality to time. This situation can be checked by the value of mean-squared displacement during the run. The ff_dif² value at t_dif is ~10 A² in the case of lithium silicates and related systems and the smaller value often means that the time scale is too short to calculate diffusion coefficients (and corresponding conductivities).

Caution is required especially for the treatment of dynamics in complex systems. If the structures have domains with different sizes, the diffusive region of MSD may not be attained easily.

Furthermore, the dynamics of ions in ionically conducting material, as well as motions of molecules and/or atoms near the glass transition temperatures, are highly heterogeneous. Namely, some ions are localized for a long time, while some other ions tend to show accelerated motions with cooperativity as shown in the following section. To obtain the slope of MSD for calculation of diffusion coefficient, time windows to average the motions should be wide enough. Many initial times will be required to obtain the mean value of heterogeneous jump motions [6].

Temperature Dependence of Characteristic Times

The next step necessary to learn the complex ionics is to represent the trend found in the MSD by more quantitative manners. As shown in previous sections, the NCL region of MSD ends at the first characteristic time t_xl, which corresponds to the onset of the jump motion of ions or particles to neighboring sites. This time-region is followed by the start of another power-law dependence of MSD after t_x2, and finally, the long-term diffusive region starting after t_dif. It is useful to examine how these time-regions behave with the changes in temperature to learn what changes of motion occur. Temperature dependences of characteristics times, as well as diffusive motions, are analyzed to give a detailed description of the dynamics of the ions. Results of the inverse of the characteristic time of Li ions in the lithium disilicate system are shown in Fig. 5.9 as a function the inverse of temperature. As already mentioned, with decreasing temperature (or increasing pressure), each time-region is expected to be longer and longer. The absolute value of the slope for 1 /tdif >^s larger than that for l/t_x2. The value for l/t_xl is the smallest but it also covers wide time scales. The slope for 1 /t_dif is comparable to that for the diffusion coefficient, D.

In other words, the time scale of t_dif determines the effective elemental process (as a mean behavior of fast and slow ions) for diffusivity. If the NCL region becomes long enough to cover the entire region of observation time, the MSD becomes flatter and the intermediate scattering function and/or other related correlation functions decay little. This situation can be well represented by the temperature dependence of the inverse of characteristic times.

In the longer time region, time development of ionic motion is modified by the correlation among successive jumps. The strong back correlation probability at lower temperatures contributes to the slowing down of the dynamics [6]. Characteristic times found in MSD are useful to judge what kind of motion is modified by the perturbation of the systems. The effect of introducing pores will be discussed in Chapter 6 by using these characteristic times.